Application Example: Measurement Uncertainty in Collimated Geometry

Determination of Activity and Measurement Uncertainty According to DIN ISO 11929

The following considerations are based on DIN ISO 11929 and its application for the evaluation of measurements in collimated geometry with experimental determination of intrinsic detector efficiency according to the Filß method.

Information about DIN ISO 11929 can be found in the section Standards – DIN ISO 11929 in Brief.

Preliminary Notes:
The starting point of the following considerations is a container (e.g., a 200 L waste container) that possibly contains gamma-emitting radioactive material of activity A . This is measured with a suitably collimated detector in an energy-resolved measurement. The container is fully scanned through suitable mechanical movements of the collimated detector and/or the container. The measurement task consists of quantifying the activity A.

The model underlying the evaluation and the relevant assumptions are described in the section Collimated Geometry - Evaluation Model According to Filß.

The activity A is the physical effect to be investigated in this case. A non-negative measurement quantity is assigned to this based on the evaluation of the measurement, which quantifies the physical effect.

Note: 
The assignment of a non-negative measurement quantity thus contains the a-priori information that there are no negative activities.

The application of DIN 11929 Part 1 for this measurement task follows according to the overview presented in section 5 (Summary of Procedures for Evaluating a Measurement and Calculating the Characteristic Limits) of the DIN standard.

Establishing the Model

The starting point is the model for counting measurements of ionizing radiation according to DIN 11929 Part 1 (Section 6.2.2) for the primary estimate y of the measurement quantity Y after inserting the estimates x_i

\[ y = G(x_1, x_2, \cdots , x_m) = (x_1 - x_2 \cdot x_3 - x_4) \cdot w = \left( \frac{n_g}{t_g} - \frac{n_0}{t_0} \cdot x_3 - x_4 \right) \cdot w \]

with

\[ w = \frac{x_6 \cdot x_8 \cdots}{x_5 \cdot x_7 \cdots} \]

and the associated standard uncertainty u(y)

\[ u(y) = \sqrt{w^2 \cdot \left[ u^2(x_1) + x_3^2 \cdot u^2(x_2) + x_2^2 \cdot u^2(x_3) + u^2(x_4) \right] + y^2 \cdot u_{rel}^2(w)} \newline = \sqrt{w^2 \cdot \left( \frac{r_g}{t_g} + x_3^2 \cdot \frac{r_0}{t_0} + r_0^2 \cdot u^2(x_3) + u^2(x_4) \right) + y^2 \cdot u_{rel}^2(w)}\]

with

\[ u_{rel} ^2 (w) = \sum_{i=5}^m \frac{u^2(x_i)}{x_i^2} \]

The size x₁ is the counting rate of the gross effect and x₂ is the counting rate of the net effect. The other input quantities x_i are calibration, correction, or influencing quantities or conversion factors.

The model for activity determination according to Filß for a completely homogeneously filled container with specific activity a

\[ A = M \cdot a = M \cdot \left[ \frac{\eta_0 \cdot A_0}{Z_0 \cdot F_0} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right)  \cdot \frac{K_2 \cdot K_3}{K_1} \right] \cdot Z \]

is transferred to this model description. For this purpose, some adjustments to Filß's model must first be made to encompass all dependencies:

1. The counting rate Z is replaced by the difference between the counting rate of the gross effect (r_g = n_g/t₀) and the counting rate of the net effect (r_r,l = n_r,l/t₀). n_g is the gross peak area determined from the measured gamma spectrum, n_r,l is the corresponding background area, and t_o is the measurement time (live time).
   \[ Z = \frac{n_g}{t_0} - \frac{n_{r,l}}{t_0} \]
   Information: More on identifying and quantifying the characteristic peaks of the measured gamma spectrum can be found in the sections Gamma Spectrometry and in the video tutorials.

    

2. The effective area F₀ of the detector's field of view is determined by the collimator and the distance S of the detector from the container surface.

    

3. The calibration quantities η₀ (Emission probability ), A₀ (Activity), Z₀ (Counting rate), and F₀ (Effective area of the detector's field of view) can be summarized in the calibration quantity H' .
   \[H^{'} = \frac{ \eta_0 \cdot A_0}{Z_0 \cdot F_0} \]
   
   Notes:

   • The determination of energy-dependent calibration quantities H' (with their corresponding uncertainties u) is only carried out at recurring time intervals or when there is a reason to verify the efficiency values. A separate consideration of the quantities included in H' can therefore be disregarded.
   • The calibration quantity H' is practically determined only for a finite number of energies that cover the range of energies to be evaluated. If necessary, a (linear) interpolation between the nearest known values of H' for the energy to be evaluated is to be performed.

    

4. The collimated detector "scans" the container completely in the segmented gamma scan measurements. It is possible that the active matrix is not within the field of view of the collimated detector during the entire measurement. This property is taken into account by the correction factor K₃. In the case of a multirotation scan or a spiral scan, K₃ can be described by the ratio of the total scan height h to the height of the active matrix h^*.

   \[K_3 = \frac{h}{h^{*}}\]

Thus, for the model according to Filß, the expression

\[ A = M \cdot a = M \cdot \left[ H^{'} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right)  \cdot \frac{K_2 }{K_1} \cdot \frac{h}{h^{*}} \right] \cdot Z \]

is obtained, and thus

\[ G(r_g, r_{r,l},M,\cdots , K_1) = A = \left( \frac{n_g}{t_0} - \frac{n_{r,l}}{t_0} \right) \cdot w \]

with

\[ w = M \cdot \left[ H^{'} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right)  \cdot \frac{K_2}{K_1}  \cdot \frac{h}{h^{*}} \right] \]

The assignment of the individual factors of this equation to the estimates for x_i can be made by simple comparison (note: any existing background (e.g., from sources in the vicinity) is not considered in this consideration (i.e., it applies: x₂ = x₃ = 0); furthermore, the dependencies of K₁ and K₂ on the linear attenuation coefficients and material thicknesses are not explicitly taken into account).

Preparation of Input Data and Specifications

In the second step, the corresponding estimates x_i and uncertainties u(x_i) for all input quantities X_i are determined and the specifications for the probabilities α, β, and γ are made.

The probabilities for type I and type II errors are often set at 5%, i.e., α = β = 0.05, as well as for the probability of the confidence interval (1 - γ), i.e., γ = 0.05. With these specifications, the quantiles of the standardized normal distribution result in k_1-α = k_1-β = 1.65 as well as k_1-γ/2 = 1.96. The values for the quantiles are tabulated.

Calculation of the Primary Measurement Result A with Standard Uncertainty ũ(Ã)

The primary measurement result, i.e., the activity A, can be determined by substituting the quantities assigned to the estimates x_i for the input quantities into the above equation for G(...) = A . Accordingly, the standard uncertainty ũ(Ã) is determined by substituting the partial derivatives and the quantities assigned to the estimates x_i with the above equation for u(y) .

Calculation of Standard Uncertainty ũ(Ã)

To determine the standard uncertainty, the true value à is needed. Since this value is not known, the non-negative primary measurement result is approximated as (à ≈ A). If this is negative, the true value is set to 0.

Since the input quantity r_g (gross counting rate) in the present case is Poisson-distributed (counting measurement) and no net effect is considered, the estimate x₁ results in

\[ r_g = \frac{\tilde{A}}{w} + r_{r,l} \]

From this, the true standard uncertainty (based on the approximation (Ã ≈ A) can be calculated.

\[ \tilde{u}(\tilde{A} ) = \sqrt{ w^2 \left[ \frac{\tilde{A}}{w} + r_{r,l} + u^2(r_{r,l}) \right] + \tilde{A}^2 \cdot u^2_{rel}(w) } \]

Calculation of the Detection Limit y^*

To determine the detection limit, the approximations (Ã ≈ A) and ũ(Ã) ≈ u(A) and the specified probability α are used. For this purpose, the determination equation is obtained

\[ A^* = k_{1 - \alpha} \cdot \tilde{u}(0) = k_{1 - \alpha} \cdot w \cdot \sqrt{r_{r,l} + u^2(r_{r,l})} \]

If the detection limit A^* is less than the determined primary measurement result A, one can assume that the physical effect is present. In the case under consideration, this would mean that the radionuclide for which the evaluation was conducted is present.

Calculation of the Quantification Limit A^#

The calculation of the detection limit A^#, the smallest true value for the activity A, is done with a specified probability β and takes into account the associated detection limit.

\[ A^{\#} = k_{1 - \alpha} \cdot \tilde{u}(0) + k_{1-\beta} \cdot \sqrt{w^2 \cdot \left[ \frac{A^{\#}}{w} + r_{r,l} + u^2(r_{r,l}) \right] + A^{\#} \cdot u^2_{rel}(w)} \]

This implicit equation for the quantification limit can be analytically determined by solving for A^#.

\[  A^{\#} = \frac {-\sqrt{ \left( 2 \cdot k_{1-\alpha} \cdot \tilde{u}(0) \right)^2 - 4 \cdot \left( k_{1-\beta}^2 \cdot u_{rel}^2(w) - 1 \right) \cdot \left( \frac{ 2 \cdot k_{1-\beta}^2 \cdot r_{r,l} \cdot w^2}{t_p} - k_{1-\alpha}^2 \cdot \tilde{u}(0)^2 \right) }} {2 \cdot \left( (k_{1-\beta})^2 \cdot  u_{rel}^2(w) - 1 \right)} - \newline - \frac{2 \cdot k_{1-\alpha} \cdot \tilde{u}(0) - \frac{k_{1-\beta}^2 \cdot w}{t_p} } {2 \cdot \left( (k_{1-\beta})^2 \cdot  u_{rel}^2(w) - 1 \right)}\]

Calculation of the Confidence Limits A^< and A^>

The calculation of the lower (A^<) and upper limit (A^>) of the confidence interval is performed using the primary measurement result A and the associated standard uncertainty ũ(Ã) for the specified probability 1-γ.

\[ A^{\triangleleft} =A - k_p \cdot u(y) \; \; \mathrm{with} \; p = w \cdot \left( 1-\frac{y}{2} \right)) \] \[ A^{\triangleright} =A + k_q \cdot u(y) \; \; \mathrm{with} \; q = 1- w \cdot \frac{y}{2} \]

with

\[ \omega = \Phi \left( \frac{A}{u(A)} \right) \]

The values of the standardized normal distribution Φ(t) are tabulated.

The true value of the measurement quantity is within the confidence interval with probability 1-γ.

Calculation of the Estimate ŷ of the Measurement Quantity with Standard Uncertainty u(ŷ)

The physical effect is considered detected when the primary measurement result A is greater than the detection limit (A > A^*).

The best estimate  of the measurement quantity is then given by

\[ \hat A = A + \frac{u(A) \cdot \exp \left\lbrace -A^2 /\left[ 2\cdot u^2(A)\right] \right\rbrace}{\omega \cdot \sqrt{2\pi}} \]

with the standard deviation

\[ u(\hat{A}) = \sqrt{u^2(A) - (\hat{A} - A) \cdot \hat{A}} \]

Preparation of the Test Report – Documentation

The conclusion of a measurement with evaluation is summarized in a complete documentation of all used data. The goal is for the evaluation to be retraced at any point in the future.

In the following exemplary structure of a test report, the gamma-spectrometric evaluations (i.e., the determination of gross peak areas and peak background) are not included. These should be recorded in a separate test report (see example). The corresponding data from the measurement should be entered in the gray fields. From these, the specifications for the empty fields are calculated. For this purpose, a test report template that automatically calculates these values is advantageous. When describing the evaluation procedure used, the references should be specified (if applicable, with revision and/or publication date).

Evaluation Procedure

Measurement in collimated geometry with evaluation model according to Filß based on DIN ISO 11929, Section 6.2.2, model for counting measurements of ionizing radiation.

Specifications

Parameters

*Additional specifications regarding other wall layers may need to be added 

Results and Characteristic Quantities

Assessment

At this point, the results are summarized and assessed once again. This should contain the following information:

1. 1. • Primary measurement result is above/below the detection limit
      • Quantification limit is above/below the specified reference value. The measurement method is suitable/not suitable as a detection method for the measurement purpose. (Alternatively: Since no reference value was provided, the assessment of the measurement method is not applicable)
      • Lower and upper confidence limits
      • Best estimate with assigned standard deviation

Auxiliaries

For the determination of characteristic quantities with the specifications and parameters, a calculation tool is available in the section Measurement Uncertainty in Collimated Geometry.